4 Evolution of energy
The deformation and failure of rock are always accompanied by energy
dissipation (Xie et al., 2005), and some researcher hold the view that
the energy dissipation can accurately reflect the change in the
mechanical properties of rocks (Ye et al., 2001). Assuming no heat
exchange between the sample and external, the work done by the external
load on the rock specimen (U ) is transformed into the elastic
strain energy (U e) and dissipated energy
(U d), and the energy equation can be expressed as
U = U e + U d
The calculation energy under triaxial cyclic loading process is shown in
Fig. 6. The elastic strain energy (U e) contained
axial strain energy () and radial strain energy (), as shown in Eq. (3).
is the area between the unloading axial stress-strain curve and the
axial strain axis, is the area the unloading confining pressure-strain
curve and the radial strain axis. The dissipated energy
(U d) is the difference between the input energy
(U ) and elastic strain energy (U e), andU d also contained axial dissipated energy () and
radial dissipated energy (), as shown in Fig. 6.
where is input axial energy, is input radial energy. is unloading axial
stress, and are unloading axial and radial strain, respectively.
Figure 7 illustrates the variation of axial input energy, radial input
energy, axial strain energy, radial strain energy, dissipated energy of
granite specimens under triaxial cyclic loading with cycle number
(σ 3 = 20MPa). It is clear that the radial input
and elastic strain energy is relatively small compared with axial input
energy, axial elastic strain energy and dissipated energy. Therefore, we
mainly discuss the axial input energy, axial elastic strain energy and
dissipated energy in this paper. Before damage threshold strength, the
axial input energy, axial elastic strain energy and dissipated energy
non-linearly increase with cycle number. When T = 25 and 300°C,
axial input energy almost transfer as axial strain energy. However,
axial strain energy and dissipated energy are near equal when T =
600°C, it indicates that damage occur in specimen once apply loading.
This characteristic corresponds well with the axial plastic strain in
Fig. 4. After the damage threshold strength, the difference between
axial input and elastic strain energy increases, and the axial elastic
strain energy reach to the peak at the peak strength. The increasing
rate of dissipated energy increases with cycle number. After the peak
strength, axial elastic strain released and macro-cracks formed, which
result in the quick increase of dissipated energy. As the strength and
elastic modulus of specimen decreases, the increasing ratio of axial
input strain decreases gradually. When the loading enter to the residual
strength stage, the axial elastic strain energy remains near constant,
while the dissipated energy is near parallel to the axial input energy,
which indicates that axial input energy almost transfer to frication
energy. However, the difference between axial energy and dissipated
energy increases with cycle number, and the radial input energy
increases obviously when T = 600°C. It means that the radial
expansion is obvious, and a part of axial input energy transfer as
radial input energy.
To investigate the effect of confining pressure on energy evolution, the
variation of axial input energy, axial elastic strain energy and
dissipated energy of granite specimen under triaxial cyclic loading at
room temperature with cycle number is illustrated in Fig. 8. It can be
seen that axial input energy and dissipated energy non-linearly
increases with cycle number on S type, as shown in Figs. 8a-b. With
increasing confining pressure, the axial input energy and its increasing
rate increase, this characteristic is more obvious between uniaxial
compression and triaxial compression when σ 3 = 10
MPa. Confining pressure increases the carrying capacity of rock, and
need more dissipated energy to destroy the rock, therefore when the
specimen failed, the dissipated energy increase with increasing
confining pressure. Confining pressure also restrain the crack
initiation and propagation, therefore the variation of dissipated energy
of specimen with confining is irregular before failure. The axial
elastic strain energy first increases and then decreases before reaching
a plateau at residual strength stage with cycle number, as shown in Fig.
8c. With increasing confining pressure, the axial elastic strain energy
value, the increasing and decreasing rate increase with increasing
confining pressure. It indicates the capacity to store strain energy
increases with increasing confining pressure. As more strain elastic
energy is stored in specimen under higher confining pressure, the
specimen destroy seriously during strain elastic energy release process,
which result in the increase of decreasing rate of axial elastic strain
energy with increasing confining pressure. Frication force between
macro-crack is larger under higher confining pressure, which results in
the larger residual strength and axial elastic strain energy.
Plastic strain and dissipated energy were widely used to define the
damage parameter of rock under cyclic loading (Eberhardt et al., 1999,
Yang et al., 2015, Wang et al., 2018). However, the relationship between
plastic strain and dissipated energy is not discussed before. Therefore,
the variation of dissipated energy of granite under triaxial cyclic
loading after different high temperature treatment with axial plastic
strain is illustrated in Fig. 9. It can be seen that dissipated energy
non-linearly increases with axial plastic strain on convex type. At
initial loading stage, there is few crack in the specimen, and need more
dissipated energy to initiate crack. As the axial plastic strain
increase, more cracks are induced in the specimen, which make the
initiation of crack easier and result in the decrease of increasing rate
of dissipated energy with axial plastic strain gradually. After the
macro-crack formed, energy mainly dissipate on the frictional slippage
between macro-crack, which result in the linear increases of it with
plastic strain. Due to more thermal cracks are induced in the specimens
when T = 600°C, the non-linear characteristic is unobvious
compare with that when T = 25 and 300°C.
It can be also seen that the dissipated energy increases with increasing
confining pressure at same axial plastic strain, it indicates that it
need more energy to initiate crack under higher confining pressure.
Confining pressure also increases friction between macro-crack, which
result in the increase of rising rate of dissipated energy with plastic
strain. With increasing temperature, more thermal cracks are induced in
the specimen, and the dissipated energy is more sensitive to confining
pressure.